Equivalence results for Cesàro submethods. (English) Zbl 0949.40015

In this paper the author defines Cesàro submethods \(C_\lambda\) as follows. Let \(E\) be the range of a strictly increasing sequence of positive integers, say \[ E=\bigl\{\lambda(n)\bigr\}^\infty_{n=1}\quad\text{and}\quad(c_\lambda x)_n= {1\over\lambda(n)} \sum^{\lambda(n)}_{k=1} x_k \] where \(\{x_k\}\) is a sequence of real or complex numbers and \(n=1,2,3,\dots,C_\lambda\) being a subsequence of the \(C_1\) method and regular for any \(\lambda\). D. H. Armitage and I. J. Maddox [A new type of Cesàro mean, Analysis 9, No. 1-2, 195-204 (1989; Zbl 0693.40009)] proved inclusion and Tauberian results for \(C_\lambda\) methods. The author here has expanded the previous results by examining further inclusion properties of the \(C_\lambda\) method for bounded sequences and its relationship to statistical convergence. His theorems are used to prove a “condensation test” for statistical convergence.
Reviewer: I.L.Sukla (Orissa)


40G05 Cesàro, Euler, Nörlund and Hausdorff methods
40D25 Inclusion and equivalence theorems in summability theory
40C05 Matrix methods for summability
40D20 Summability and bounded fields of methods


Zbl 0693.40009
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